Context-free grammar for all words not of the form w#w

Question

I was asked to define a CFG for the complement of $\{w\#w \mid w \in \{0,1\}^*\}$ and I'm struggling to define it. I think it is quite similar to defining a CFG for the complement of $\{ww \mid w \in \{0,1\}^*\}$, but I'm finding that the '#' really messes things up. Is the language even context-free?

Answer 1

If a word is not of the form $w\#w$ then one of the following must happen:

• It contains no $\#$.
• It contains more than one $\#$.
• It is of the form $x\#y$, where $x,y \in \{0,1\}^*$ and $x \neq y$, in which case one of the following must happen:
  • $|x| > |y|$ or $|x| < |y|$.
  • $x_i = 0$ and $y_i = 1$ for some $i$, or vice versa.

The only case which is non-trivial to handle is the last one, in which case we have a word of the form $$ x_1 0 x_2 \# y_1 1 y_2, $$ where $|x_1| = |y_1|$. We can split this into $$ x_1 \;\; 0x_2\# \;\; y_1 \;\; 1y_2, $$ where $|x_1| = |y_1|$. This should make it clear how to construct a CFG for such words.

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Altogether, we obtain the following grammar: \begin{align} &S \to A \mid B \mid C \mid D \mid E \mid F \\ &A \to 0A \mid 1A \mid \epsilon \\ &B \to A\#B \mid A\#A\#A \\ &C \to A0G \mid A1G \\ &D \to G0A \mid G1A \\ &G \to 0G0 \mid 0G1 \mid 1G0 \mid 1G1 \mid \# \\ &E \to E_11A \\ &E_1 \to 0E_10 \mid 0E_11 \mid 1E_10 \mid 1E_11 \mid 0A\# \\ &F \to F_10A \\ &F_1 \to 0F_10 \mid 0F_11 \mid 1F_10 \mid 1F_11 \mid 1A\# \\ \end{align}

In this grammar:

• $A$ generates words without $\#$
• $B$ generates words with more than one $\#$
• $C$ generates words of the form $x\#y$ with $|x| > |y|$
• $D$ generates words of the form $x\#y$ with $|x| < |y|$
• $E$ generates words of the form $x_10x_2\#y_11y_2$ with $|x_1|=|y_1|$
• $F$ generates words of the form $x_11x_2\#y_10y_2$ with $|x_1|=|y_1|$
• $G$ generates words of the form $x\#y$ with $|x| = |y|$